A decomposition of the natural embedding spaces for the symplectic dual polar spaces Original Research Article

Let e be the Grassmann-embedding of the symplectic dual polar space image into PG(W), where W is a image-dimensional vector space over image. For every point z of image and every image, Δi(z) denotes the set of points at distance i from z. We show that for every pair {x,y} of mutually opposite points of image can be written as a direct sum W0circled plusW1circled pluscdots, three dots, centeredcircled plusWn such that the following four properties hold for every iset membership, variant{0,…,n}: (1) left angle brackete(Δi(x)∩Δn-i(y))right-pointing angle bracket=PG(Wi); (2) left angle brackete(union operatorjless-than-or-equals, slantiΔj(x))right-pointing angle bracket=PG(W0circled plusW1circled pluscdots, three dots, centeredcircled plusWi); (3) left angle brackete(union operatorjless-than-or-equals, slantiΔj(y))right-pointing angle bracket=PG(Wn-icircled plusWn-i+1circled pluscdots, three dots, centeredcircled plusWn); (4) image.